Factor out known multipliers in algebraic expressions?

I have this function of a[t], and it's a long function of other variables in time and some constant parameter (Ixx, Iyy, Izz), which I really want to explicit in a somatory of any combination of those constant parameters.
I've tried to Simplify and then Apart, but Mathematica doesn't group similar coefficients. I wonder if I could get this result applying some kind of mapping (...)?

Summarizing, what I want is a way to say : rewrite a[t] factorizing in a sum by these terms (Ixx, Iyy, Izz) on any combination between them.
( In other way, I could ask the same by rewriting a[t] factorizing in terms that don't vary in time? Would this be possible/easier? )

works in your simple example, but for sure it will get tedious in more complicated cases.

And what is wrong with the method I mentioned first? This gives num1 as a polynomial in Ixx, Ixx^2, Ixx * Iyy and so on with coefficients as functions depending only of t. Same for the denominator and then you can write your a [ t ] in the form you want.